book 2007/8 page 1 Copyright ©2007 by the Society for Industrial and Applied Mathematics. This electronic version is for personal use and may not be duplicated or distributed. Chapter 6 PID Controller Design PID (proportional integral derivative) control is one of the earlier control strategies [59]. Its early implementation was in pneumatic devices, followed by vacuum and solid state analog electronics, before arriving at today’s digital implementation of microprocessors. It has a simple control structure which was understood by plant operators and which they found relatively easy to tune. Since many control systems using PID control have proved satisfactory, it still has a wide range of applications in industrial control. According to a survey for process control systems conducted in 1989, more than 90 of the control loops were of the PID type [60]. PID control has been an active research topic for many years;see the monographs [60–64]. Since many process plants controlled by PID controllers have similar dynamics it has been found possible to set satisfactory controller parameters from less plant information than a complete mathematical model. These techniques came about because of the desire to adjust controller parameters in situ with a minimum of effort, and also because of the possible difficulty and poor cost benefit of obtaining mathematical models. The two most popular PID techniques were the step reaction curve experiment, and a closed-loop “cycling” experiment under proportional control around the nominal operating point. In this chapter, several useful PID-type controller design techniques will be presented, and implementation issues for the algorithms will also be discussed. In Sec. 6.1, the pro- portional, integral, and derivative actions are explained in detail, and some variations of the typical PID structure are also introduced. In Sec. 6.2, the well-known empirical Ziegler– Nichols tuning formula and modified versions will be covered. Approaches for identifying the equivalent first-order plus dead time model, which is essential in some of the PID con- troller design algorithms, will be presented. A modified Ziegler–Nichols algorithm is also given. Some other simple PID setting formulae such as the Chien–Hrones–Reswick for- mula, Cohen–Coon formula, refined Ziegler–Nichols tuning, Wang–Juang–Chan formula and Zhuang–Atherton optimum PID controller will be presented in Sec. 6.3. In Sec. 6.4, the PID tuning formulae for FOIPDT (first- order lag and integrator plus dead time) and IPDT (integrator plus dead time) plant models, rather than the FOPDT (first-order plus dead time) model, will be given. A graphical user interface (GUI) implementing hundreds of PID controllers tuning formulae for FOPDT model will be given in Sec. 6.5. In Sec. 6.6, an uncorrectedoptimization-based design algorithm, together with a GUI for optimal proofs controller design, is 183 From "Linear Feedback Control" by Dingyu Xue, YangQuan Chen, and Derek P. Atherton. This book is available for purchase at www.siam.org/catalog. book 2007/8/ page 18 Copyright ©2007 by the Society for Industrial and Applied Mathematics. This electronic version is for personal use and may not be duplicated or distributed. 184 Chapter 6. PID Controller Design given. In Sec. 6.7, some of the advanced topics on PID control will be presented, such as integrator windup phenomenon and prevention, and automatic tuning techniques. Finally, some suggestions on controller structure selections for practical process control are pro- vided. 6.1 Introduction 6.1.1 The PID Actions A typical structure of a PID control system is shown in Fig. 6.1, where it can be seen that in a PID controller, the error signal e(t) is used to generate the proportional, integral, and derivative actions, with the resulting signals weighted and summed to form the control signal u(t) applied to the plant model. A mathematical description of the PID controller is 1 t de(t) u(t) = Kp e(t) + e(τ)dτ + Td , (6.1) Ti 0 dt where u(t) is the input signal to the plant model, the error signal e(t) is defined as e(t) = r(t) − y(t), and r(t) is the reference input signal. The behavior of the proportional, integral, and derivative actions will be demonstrated individually through the following example. Example 6.1. Consider a third-order plant model given by G(s) = 1/(s + 1)3. If a pro- portional control strategy is selected, i.e., Ti →∞and Td → 0 in the PID control strategy, for different values of Kp, the closed-loop responses of the system can be obtained using the following MATLAB statements: >> G=tf(1,[1,3,3,1]); for Kp=[0.1:0.1:1], G_c=feedback(Kp*G,1); step(G_c), hold on; end figure; rlocus(G,[0,15]) The closed-loop step responses are obtained as shown in Fig. 6.2(a), and it can be seen that when Kp increases, the response speed of the system increases, the overshoot of the closed-loop system increases, and the steady-state error decreases. However when Kp is large enough, the closed-loop system becomes unstable, which can be directly concluded from the root locus analysis in Sec. 3.4. The root locus of the example system is shown ............................................. PID controller . disturbance d(t) . r(t) e(t) . 6 . u(t) ?? y(t) - -. - - . - - plant - . controller . um . model 6 . ? . ........................................... measure- ment noise uncorrectedFigure 6.1. A typical PID control structure. proofs From "Linear Feedback Control" by Dingyu Xue, YangQuan Chen, and Derek P. Atherton. This book is available for purchase at www.siam.org/catalog. book 2007/8 page 1 Copyright ©2007 by the Society for Industrial and Applied Mathematics. This electronic version is for personal use and may not be duplicated or distributed. 6.1. Introduction 185 Step Response Root Locus 2.5 ← K = 1 p 2 0.5 1.5 System: G Gain: 8 1 Pole: 2.36e−005 + 1.73i 0.4 Damping: −1.37e−005 Overshoot (%): 100 0.5 Frequency (rad/sec): 1.73 0.3 0 Amplitude Imaginary Axis −0.5 0.2 −1 = −1.5 0.1 Kp 0.1 −2 0 −2.5 0 5 10 15 −2.5 −2 −1.5 −1 −0.5 0 0.5 Time (sec) Real Axis (a) closed-loop step response (b) root locus Figure 6.2. Closed-loop step responses. Step Response Step Response 2 1.6 ← = 1.8 ← = Td 0.1 Ti 0.7 1.4 1.6 1.2 ← T =1.5 1.4 d 1 1.2 1 0.8 Amplitude Amplitude 0.8 0.6 0.6 ← = 0.4 0.4 Ti 1.5 0.2 0.2 0 0 0 5 10 15 20 0 5 10 15 20 Time (sec) Time (sec) (a) PI control (b) PID control Figure 6.3. Closed-loop step responses. in Fig. 6.2(b), where it is seen that when Kp is outside the range of (0, 8), the closed-loop system becomes unstable. If we fix Kp = 1 and apply a PI (proportional plus integral) control strategy for different values of Ti, we can use the following MATLAB statements: >> Kp=1; s=tf(’s’); for Ti=[0.7:0.1:1.5] Gc=Kp*(1+1/Ti/s); G_c=feedback(G*Gc,1); step(G_c), hold on end to generate the closed-loop step responses of the example system shown in Fig. 6.3(a). The most important feature of a PI controller is that there is no steady-state error in the step response if the closed-loop system is stable. Further examination shows that if Ti is smaller than 0.6, the closed-loop system will not be stable. It can be seen that when Ti increases, uncorrectedthe overshoot tends to be smaller, but the speed of response tends toproofs be slower. From "Linear Feedback Control" by Dingyu Xue, YangQuan Chen, and Derek P. Atherton. This book is available for purchase at www.siam.org/catalog. book 2007/8 page 1 Copyright ©2007 by the Society for Industrial and Applied Mathematics. This electronic version is for personal use and may not be duplicated or distributed. 186 Chapter 6. PID Controller Design Fixing both Kp and Ti at 1, i.e., Ti = Kp = 1, when the PID control strategy is used, with different Td, we can use the MATLAB statements >> Kp=1; Ti=1; s=tf(’s’); for Td=[0.1:0.2:2] Gc=Kp*(1+1/Ti/s+Td*s); G_c=feedback(G*Gc,1); step(G_c), hold on end to get the closed-loop step response shown in Fig. 6.3(b). Clearly, when Td increases the response has a smaller overshoot with a slightly slower rise time but similar settling time. In practical applications, the pure derivative action is never used, due to the “derivative kick” produced in the control signal for a step input, and to the undesirable noise amplifica- tion. It is usually replaced by a first-order low pass filter. Thus, the Laplace transformation representation of the approximate PID controller can be written as 1 sTd U(s) = Kp 1 + + E(s). (6.2) Tis + Td 1 s N The effect of N is illustrated through the following example. Example 6.2. Consider the plant model in Example 6.1. The PID controller parameters are Kp = 1,Ti = 1, and Td = 1. With different selections of N, we can use the MATLAB commands >> Td=1; Gc=Kp*(1+1/Ti/s+Td*s); step(feedback(G*Gc,1)), hold on for N=[100,1000,10000,1:10] Gc=Kp*(1+1/Ti/s+Td*s/(1+Td*s/N)); G_c=feedback(G*Gc,1); step(G_c) end figure; [y,t]=step(G_c); err=1-y; plot(t,err) to get the closed-loop step response with the approximate derivative terms as shown in Fig.